In particle theory, the **skyrmion** ( /ˈskɜːrmi.ɒn/ ) is a topologically stable field configuration of a certain class of non-linear sigma models. It was originally proposed as a model of the nucleon by Tony Skyrme in 1961.^{ [1] }^{ [2] }^{ [3] }^{ [4] } As a topological soliton in the pion field, it has the remarkable property of being able to model, with reasonable accuracy, multiple low-energy properties of the nucleon, simply by fixing the nucleon radius. It has since found application in solid-state physics, as well as having ties to certain areas of string theory.

- Topological soliton
- Lagrangian
- Noether current
- Magnetic materials/data storage
- See also
- References
- Further reading

Skyrmions as topological objects are important in solid-state physics, especially in the emerging technology of spintronics. A two-dimensional magnetic skyrmion, as a topological object, is formed, e.g., from a 3D effective-spin "hedgehog" (in the field of micromagnetics: out of a so-called "Bloch point" singularity of homotopy degree +1) by a stereographic projection, whereby the positive north-pole spin is mapped onto a far-off edge circle of a 2D-disk, while the negative south-pole spin is mapped onto the center of the disk. In a spinor field such as for example photonic or polariton fluids the skyrmion topology corresponds to a full Poincaré beam^{ [5] } (which is, a quantum vortex of spin comprising all the states of polarization).^{ [6] }

Skyrmions have been reported, but not conclusively proven, to be in Bose–Einstein condensates,^{ [7] } thin magnetic films^{ [8] } and in chiral nematic liquid crystals.^{ [9] }

As a model of the nucleon, the topological stability of the skyrmion can be interpreted as a statement that the baryon number is conserved; i.e. that the proton does not decay. The Skyrme Lagrangian is essentially a one-parameter model of the nucleon. Fixing the parameter fixes the proton radius, and also fixes all other low-energy properties, which appear to be correct to about 30%. It is this predictive power of the model that makes it so appealing as a model of the nucleon.

Hollowed-out skyrmions form the basis for the chiral bag model (Cheshire Cat model) of the nucleon. Exact results for the duality between the fermion spectrum and the topological winding number of the non-linear sigma model have been obtained by Dan Freed. This can be interpreted as a foundation for the duality between a QCD description of the nucleon (but consisting only of quarks, and without gluons) and the Skyrme model for the nucleon.

The skyrmion can be quantized to form a quantum superposition of baryons and resonance states.^{ [10] } It could be predicted from some nuclear matter properties.^{ [11] }

In field theory, skyrmions are homotopically non-trivial classical solutions of a nonlinear sigma model with a non-trivial target manifold topology – hence, they are topological solitons. An example occurs in chiral models ^{ [12] } of mesons, where the target manifold is a homogeneous space of the structure group

where SU(*N*)_{L} and SU(*N*)_{R} are the left and right chiral symmetries, and SU(*N*)_{diag} is the diagonal subgroup. In nuclear physics, for *N* = 2, the chiral symmetries are understood to be the isospin symmetry of the nucleon. For *N* = 3, the isoflavor symmetry between the up, down and strange quarks is more broken, and the skyrmion models are less successful or accurate.

If spacetime has the topology S^{3}×**R**, then classical configurations can be classified by an integral winding number ^{ [13] } because the third homotopy group

is equivalent to the ring of integers, with the congruence sign referring to homeomorphism.

A topological term can be added to the chiral Lagrangian, whose integral depends only upon the homotopy class; this results in superselection sectors in the quantised model. In (1 + 1)-dimensional spacetime, a skyrmion can be approximated by a soliton of the Sine–Gordon equation; after quantisation by the Bethe ansatz or otherwise, it turns into a fermion interacting according to the massive Thirring model.

The Lagrangian for the skyrmion, as written for the original chiral SU(2) effective Lagrangian of the nucleon-nucleon interaction (in (3 + 1)-dimensional spacetime), can be written as

where , , are the isospin Pauli matricies, is the Lie bracket commutator, and tr is the matrix trace. The meson field (pion field, up to a dimensional factor) at spacetime coordinate is given by . A broad review of the geometric interpretation of is presented in the article on sigma models.

When written this way, the is clearly an element of the Lie group SU(2), and an element of the Lie algebra su(2). The pion field can be understood abstractly to be a section of the tangent bundle of the principal fiber bundle of SU(2) over spacetime. This abstract interpretation is characteristic of all non-linear sigma models.

The first term, is just an unusual way of writing the quadratic term of the non-linear sigma model; it reduces to . When used as a model of the nucleon, one writes

with the dimensional factor of being the pion decay constant. (In 1 + 1 dimensions, this constant is not dimensional and can thus be absorbed into the field definition.)

The second term establishes the characteristic size of the lowest-energy soliton solution; it determines the effective radius of the soliton. As a model of the nucleon, it is normally adjusted so as to give the correct radius for the proton; once this is done, other low-energy properties of the nucleon are automatically fixed, to within about 30% accuracy. It is this result, of tying together what would otherwise be independent parameters, and doing so fairly accurately, that makes the Skyrme model of the nucleon so appealing and interesting. Thus, for example, constant in the quartic term is interpreted as the vector-pion coupling ρ–π–π between the rho meson (the nuclear vector meson) and the pion; the skyrmion relates the value of this constant to the baryon radius.

The local winding number density is given by

where is the totally antisymmetric Levi-Civita symbol (equivalently, the Hodge star, in this context).

As a physical quantity, this can be interpreted as the baryon current; it is conserved: , and the conservation follows as a Noether current for the chiral symmetry.

The corresponding charge is the baryon number:

As a conserved charge, it is time-independent: , the physical interpretation of which is that protons do not decay.

In the chiral bag model, one cuts a hole out of the center and fills it with quarks. Despite this obvious "hackery", the total baryon number is conserved: the missing charge from the hole is exactly compensated by the spectral asymmetry of the vacuum fermions inside the bag.^{ [14] }^{ [15] }^{ [16] }

One particular form of skyrmions is magnetic skyrmions, found in magnetic materials that exhibit spiral magnetism due to the Dzyaloshinskii–Moriya interaction, double-exchange mechanism ^{ [17] } or competing Heisenberg exchange interactions.^{ [18] } They form "domains" as small as 1 nm (e.g. in Fe on Ir(111)).^{ [19] } The small size and low energy consumption of magnetic skyrmions make them a good candidate for future data-storage solutions and other spintronics devices.^{ [20] }^{ [21] }^{ [22] } Researchers could read and write skyrmions using scanning tunneling microscopy.^{ [23] }^{ [24] } The topological charge, representing the existence and non-existence of skyrmions, can represent the bit states "1" and "0". Room-temperature skyrmions were reported.^{ [25] }^{ [26] }

Skyrmions operate at current densities that are several orders of magnitude weaker than conventional magnetic devices. In 2015 a practical way to create and access magnetic skyrmions under ambient room-temperature conditions was announced. The device used arrays of magnetized cobalt disks as artificial Bloch skyrmion lattices atop a thin film of cobalt and palladium. Asymmetric magnetic nanodots were patterned with controlled circularity on an underlayer with perpendicular magnetic anisotropy (PMA). Polarity is controlled by a tailored magnetic-field sequence and demonstrated in magnetometry measurements. The vortex structure is imprinted into the underlayer's interfacial region by suppressing the PMA by a critical ion-irradiation step. The lattices are identified with polarized neutron reflectometry and have been confirmed by magnetoresistance measurements.^{ [27] }^{ [28] }

A recent (2019) stydy^{ [29] } demonstrated a way to move skyrmions, purely using electric field (in the absence of electric current). The authors used Co/Ni multilayers with a thickness slope and Dzyaloshinskii–Moriya interaction and demonstrated skyrmions. They showed that the displacement and velocity depended directly on the applied voltage.^{ [30] }

In 2020, a team of researchers from the Swiss Federal Laboratories for Materials Science and Technology (Empa) has succeeded for the first time in producing a tunable multilayer system in which two different types of skyrmions – the future bits for "0" and "1" – can exist at room temperature.^{[ citation needed ]}

- Hopfion, 3D counterpart of skyrmions

In particle physics, a **fermion** is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, spin 3/2, etc. These particles obey the Pauli exclusion principle. Fermions include all quarks and leptons, as well as all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics.

The **muon** is an elementary particle similar to the electron, with an electric charge of −1 *e* and a spin of 1/2, but with a much greater mass. It is classified as a lepton. As with other leptons, the muon is not known to have any sub-structure – that is, it is not thought to be composed of any simpler particles.

In chemistry and physics, a **nucleon** is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines an isotope's mass number.

In particle physics, **proton decay** is a hypothetical form of particle decay in which the proton decays into lighter subatomic particles, such as a neutral pion and a positron. The proton decay hypothesis was first formulated by Andrei Sakharov in 1967. Despite significant experimental effort, proton decay has never been observed. If it does decay via a positron, the proton's half-life is constrained to be at least 1.67×10^{34} years.

In theoretical physics, **quantum chromodynamics** (**QCD**) is the theory of the strong interaction between quarks and gluons, the fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of quantum field theory called a non-abelian gauge theory, with symmetry group SU(3). The QCD analog of electric charge is a property called *color*. Gluons are the force carrier of the theory, just as photons are for the electromagnetic force in quantum electrodynamics. The theory is an important part of the Standard Model of particle physics. A large body of experimental evidence for QCD has been gathered over the years.

In particle physics, a **pion** is any of three subatomic particles: ^{}π^{0}_{}, ^{}π^{+}_{}, and ^{}π^{−}_{}. Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more generally, the lightest hadrons. They are unstable, with the charged pions ^{}π^{+}_{} and ^{}π^{−}_{} decaying after a mean lifetime of 26.033 nanoseconds, and the neutral pion ^{}π^{0}_{} decaying after a much shorter lifetime of 84 attoseconds. Charged pions most often decay into muons and muon neutrinos, while neutral pions generally decay into gamma rays.

The **Standard Model** of particle physics is the theory describing three of the four known fundamental forces in the universe, as well as classifying all known elementary particles. It was developed in stages throughout the latter half of the 20th century, through the work of many scientists around the world, with the current formulation being finalized in the mid-1970s upon experimental confirmation of the existence of quarks. Since then, confirmation of the top quark (1995), the tau neutrino (2000), and the Higgs boson (2012) have added further credence to the Standard Model. In addition, the Standard Model has predicted various properties of weak neutral currents and the W and Z bosons with great accuracy.

In theoretical physics, a **chiral anomaly** is the anomalous nonconservation of a chiral current. In everyday terms, it is equivalent to a sealed box that contained equal numbers of left and right-handed bolts, but when opened was found to have more left than right, or vice versa.

In physics, the **Kondo effect** describes the scattering of conduction electrons in a metal due to magnetic impurities, resulting in a characteristic change in electrical resistivity with temperature. The effect was first described by Jun Kondo, who applied third-order perturbation theory to the problem to account for s-d electron scattering. Kondo's model predicted that the scattering rate of conduction electrons off the magnetic impurity should diverge as the temperature approaches 0 K. Extended to a lattice of *magnetic impurities*, the Kondo effect likely explains the formation of *heavy fermions* and *Kondo insulators* in intermetallic compounds, especially those involving rare earth elements like cerium, praseodymium, and ytterbium, and actinide elements like uranium. The Kondo effect has also been observed in quantum dot systems.

A **metamaterial** is any material engineered to have a property that is not found in naturally occurring materials. They are made from assemblies of multiple elements fashioned from composite materials such as metals and plastics. The materials are usually arranged in repeating patterns, at scales that are smaller than the wavelengths of the phenomena they influence. Metamaterials derive their properties not from the properties of the base materials, but from their newly designed structures. Their precise shape, geometry, size, orientation and arrangement gives them their smart properties capable of manipulating electromagnetic waves: by blocking, absorbing, enhancing, or bending waves, to achieve benefits that go beyond what is possible with conventional materials.

In nuclear physics, the **chiral model**, introduced by Feza Gürsey in 1960, is a phenomenological model describing effective interactions of mesons in the chiral limit, but without necessarily mentioning quarks at all. It is a nonlinear sigma model with the principal homogeneous space of the Lie group SU(*N*) as its target manifold, where *N* is the number of quark flavors. The Riemannian metric of the target manifold is given by a positive constant multiplied by the Killing form acting upon the Maurer-Cartan form of SU(*N*).

The **QCD vacuum** is the vacuum state of quantum chromodynamics (QCD). It is an example of a *non-perturbative* vacuum state, characterized by non-vanishing condensates such as the gluon condensate and the quark condensate in the complete theory which includes quarks. The presence of these condensates characterizes the **confined phase** of quark matter.

This article describes the mathematics of the **Standard Model** of particle physics, a gauge quantum field theory containing the internal symmetries of the unitary product group SU(3) × SU(2) × U(1). The theory is commonly viewed as containing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs boson.

**Chiral perturbation theory** (ChPT) is an effective field theory constructed with a Lagrangian consistent with the (approximate) chiral symmetry of quantum chromodynamics (QCD), as well as the other symmetries of parity and charge conjugation. ChPT is a theory which allows one to study the low-energy dynamics of QCD on the basis of this underlying chiral symmetry.

In particle physics, **chiral symmetry breaking** is the spontaneous symmetry breaking of a chiral symmetry – usually by a gauge theory such as quantum chromodynamics, the quantum field theory of the strong interaction. Yoichiro Nambu was awarded the 2008 Nobel prize in physics for describing this phenomenon.

** N = 4 supersymmetric Yang–Mills** (

The **proton magnetic moment** is the magnetic dipole moment of the proton, symbol *μ*_{p}. Protons and neutrons, both nucleons, comprise the nucleus of an atom, and both nucleons act as small magnets whose strength is measured by their magnetic moments. The magnitude of the proton's magnetic moment indicates that the proton is not an elementary particle.

In physics, **magnetic skyrmions** are quasiparticles which have been predicted theoretically and observed experimentally in condensed matter systems. Skyrmions, named after British physicist Tony Hilton Royle Skyrme, can be formed in magnetic materials in their 'bulk' such as in MnSi, or in magnetic thin films. They can be achiral, or chiral in nature, and may exist both as dynamic excitations or stable or metastable states. Although the broad lines defining magnetic skyrmions have been established de facto, there exist a variety of interpretations with subtle differences.

The term **Dirac matter** refers to a class of condensed matter systems which can be effectively described by the Dirac equation. Even though the Dirac equation itself was formulated for fermions, the quasi-particles present within Dirac matter can be of any statistics. As a consequence, Dirac matter can be distinguished in fermionic, bosonic or anyonic Dirac matter. Prominent examples of Dirac matter are Graphene, topological insulators, Dirac semimetals, Weyl semimetals, various high-temperature superconductors with -wave pairing and liquid Helium-3. The effective theory of such systems is classified by a specific choice of the Dirac mass, the Dirac velocity, the Dirac matrices and the space-time curvature. The universal treatment of the class of Dirac matter in terms of an effective theory leads to a common features with respect to the density of states, the heat capacity and impurity scattering.

A **hopfion** is a topological soliton. It is a stable three-dimensional localised configuration of a three-component field with a knotted topological structure. They are the three-dimensional counterparts of skyrmions, which exhibit similar topoligical properties in 2D.

- ↑ Skyrme, T. H. R.; Schonland, Basil Ferdinand Jamieson (1961-02-07). "A non-linear field theory".
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- ↑ The same classification applies to the mentioned effective-spin "hedgehog" singularity": spin upwards at the northpole, but downward at the southpole.

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- Developments in Magnetic Skyrmions Come in Bunches, IEEE Spectrum 2015 web article

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